S.G.
Mu daughter was exactly like that when she was one year old. I couldn't do anything around the house when I was at home with her, not even make myself coffee in the morning. If turn around and step away she starts to cry. But she loved playing with me or reading a book, or sometimes she would play by herself fine and I was just sitting in the corner watching her, and I wa thinking I could just quietly get up and go to the kitchen, but as soon as I get up she notices and stands up and goes to me and doesn't want to let me go anywhere. Well, I still needed to do things around the house, so every time I has to go to another room I would just pick her up with one hand and carry her with me, and I had my other hand free to do stuff that I wanted to do. It's was a little difficult for me, but manageable.
Now my daughter is 2 and I don't have tat problem anymore. She is more independant, and can play in the living room and push her car around while I am cleaning up in the kitchen. I can also leave her in her room and tell her to wait for me, and I would go to another room to pick up the phone (or whatever), and return shortly, and she would be waiting for me without crying.
I think 1 year is just such an age that they always want to be around you and feel safe :)